document.write( "Question 1183949: Solve 2x^3 + 3x^2 +hx + k = 0 and find the values of h and k, given that -3 is the first root and the third root is twice the second. \n" ); document.write( "

Algebra.Com's Answer #814462 by ikleyn(52799) $\"\"$ $\"About$

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\n" ); document.write( "Solve 2x^3 + 3x^2 +hx + k = 0 and find the values of h and k, given that -3 is the first root and the third root is twice the second.
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document.write( "Let x be the second root.  \r\n" );
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document.write( "Then the third root is 2x.\r\n" );
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document.write( "Next, according to Vieta's theorem, the ratio of the coefficient at  x^2  to the leading coefficient,\r\n" );
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document.write( "i.e. the number  ,  is the sum of the three roots of the equation, taken with the opposite sign\r\n" );
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document.write( "     = - .\r\n" );
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document.write( "Simplify and find x\r\n" );
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document.write( "    3 = -2*((-3) + 3x)\r\n" );
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document.write( "    3 = 6 - 6x\r\n" );
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document.write( "    6x = 6 - 3\r\n" );
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document.write( "    6x = 3\r\n" );
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document.write( "    x = 3/6 = 1/2.\r\n" );
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document.write( "Thus, the second root is  1/2;   the third root is 1.\r\n" );
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document.write( "According to Vieta's theorem,  the ratio    is the product of the three roots taken with the opposite sign\r\n" );
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document.write( "     = -  = 3/2;  hence,  k = 3.\r\n" );
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document.write( "According to Vieta's theorem, the ratio    is the sum of in-pairs product of the roots\r\n" );
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document.write( "     =  =  = -4;  hence, h = -8\r\n" );
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document.write( "ANSWER.  k = 3;  h = -8;  the second root is  ;  the third root is 1.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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